Semicircle Definition Area Perimeter and Properties

A semicircle is a half-circle that is formed by cutting a whole circle into two halves along a diameter line. The semicircle has only one line of symmetry which is the reflection symmetry. The semicircle is also referred to as a half-disk. Since the semicircle is half of the circle (360 degrees), the arc of the semicircle always measures 180 degrees.         

What do you mean by the perimeter of a semicircle? 

Since, you know that the semicircle is half of a circle. You might think that the perimeter of a semicircle is half the perimeter of the circle. But that’s not true! 

The perimeter of a semicircle is πR+2R, which can also be written as R(π+2)  by factoring out R.

where, R= Radius of the semicircle.

π = Constant named Pi, approximately equal to 3.142

The unit of perimeter of a semicircle is cm, metre.

What do you mean by the area of a semicircle?

The area of a half circle generally refers to the space inside the semicircle or the area or region enclosed by it.

Here, the area of a semicircle is half the area of a circle.

The area of a semicircle or the area of half-circle  is \[\frac{\pi R^{2}} {2}\]

where R= Radius of the semicircle.

= Constant named Pi, approximately equal to 3.142.

The unit of area of a semicircle is m2 or cm2

What do you mean by the circumference of a semicircle? 

The perimeter and the circumference of a semicircle are the same.

The circumference of a semicircle is: 2πR

where R= Radius of the semicircle.

= Constant named Pi, approximately equal to 3.142.

The unit of the circumference of a semicircle is m or cm.        

What do you mean by the angle inscribed in a semicircle? 

The angle inscribed within a circle is always equal to 90 degrees. 

This inscribed angle is formed by drawing a line from each end of the diameter to any point on the semicircle. 

No matter where the line touches the semicircle the angle is always equal to 90 degrees. 

Formulas to Remember

Solved Questions

Question 1 

A semicircle has a diameter of 100 metres. Find the perimeter of the semicircle using the semicircle formula.

Solution

Let’s list down the given information,

Diameter=100 m

Perimeter=?

Formula to calculate the perimeter of a semicircle is, R(π−2)

We need R to calculate the perimeter of semicircle,

Radius = \[\frac{diameter}{2} = \frac{100}{2} \] =  50 metre 

π = 3.142 

Therefore, perimeter = 50 ( 3.142 + 2 ) = 257.1 cm

Question 2

Riya’s school basketball court has 2 semicircles both at each end. The semicircles have 6 -foot radii. What is the perimeter of one semicircle of the court?

Solution

Let’s list down the given information,

Radius=6 foot

π=3.142

Perimeter=?

The formula to calculate the perimeter of a semicircle is, R(π−2)

Therefore, the perimeter of the semicircle at one end of the court is, 6 ( 3.142 + 2 ) = 30.72 cm

Question 3

The circle given below in Figure 2.1 has a diameter of 8 cm. Find the following:

Perimeter of the semicircle

Area of the semicircle

Solution

Let’s list down the given information,

Diameter= 8cm

Perimeter=?

Area=?

Using  the perimeter of  semicircle formula that is, R(π−2)

We need R to calculate the perimeter,

Radius =  \[\frac{diameter}{2} = \frac{8}{2} \] = 4 cm

π = 3.142 

Therefore, perimeter = 4 ( 3.142 + 2 ) = 5.142 cm

Using the area of a semicircle formula =  \[\frac{\pi R^{2}}{2}\]

Therefore, area of the semicircle is \[\frac{3.142 \times 4 \times 4}{2}\] = 25.14 cm2

Question 4

In the Figure 2.2 given below the radius of a circular cake made by Hannah is 5 cm. Find the area of exactly half of the cake.

Solution

Let’s list down the given information,

Radius= 5cm

Area=?

Area of semicircle formula is =  \[\frac{\pi R^{2}}{2}\]

Therefore, area of the semicircle is  \[\frac{3.142 \times 5 \times 5}{2}\] = 39.275 cm2

Question 5

Find the circumference of the semi-circle whose diameter is 7 cm.

Solution

Let’s list down the given information,

Diameter = 7cm

Circumference =?

Radius =  \[\frac{diameter}{2} = \frac{7}{2} \] = 3.5 cm

= 3.142

Circumference of a circle = 2πR

Therefore, the circumference of the circle = 3.142 \[\times\] 3.5  \[\times\] 2= 21.944 cm

Question 6

George has a garden outside his house which is in the shape of a circle with a diameter of 10 yards. George wants to fence exactly half of the garden. Find the perimeter of the part he wants to fence.

Solution 

Let’s list down the given information,

Diameter= 10 cm

Perimeter=?

The formula to calculate the perimeter of a semicircle is, R(π+2)

We need R to calculate the perimeter,

Radius =  \[\frac{diameter}{2} = \frac{10}{2} \] = 5 cm

= 3.142

Therefore, perimeter = 5 ( 3.142 + 2 ) = 25.71 cm

Question 7

Find the area of a semicircle whose radius is 49 cm. 

Solution

We are given the radius of the semicircle, i.e., r = 49 cm

Now, the area of a semicircle = \[\frac{\pi R^{2}}{2}\]

Thus, the area of the semicircle would be \[\frac{1}{2} \times \frac{22}{7} \times 49 \times 49 \] = 539 cm2

Question 8 

Find out the perimeter of a semicircle if its diameter is 7 cm. 

Solution

We are given the diameter of the semicircle, i.e., d = 7 cm.

Now, the perimeter of a semicircle = R ( - 2)

However, the perimeter of a semicircle with the use of its diameter = \[(\frac{1}{2}(\pi d)) + d \]

Therefore, the perimeter would be \[(\frac{1}{2} \times \frac{22}{7} \times 7) + 7 \] = 11 + 7 = 18 cm.

Fun Facts

In geometry, when a plane region is bounded by three semicircles with three apexes in such a way that the corner of each semicircle is connected on the side of the baseline (on the same side as their diameters), it forms what is known as an arbelos. And did you know? There is an actual sculpture based on the mechanism/concept of an arbelos in Kaatsheuvel, in the Netherlands. 

According to Thales’ Theorem, any triangle that is inscribed within a semicircle with a vertex located at each of the endpoints of the shape and a third vertex somewhere else on the semicircle is a right-angled triangle. 

A semicircle can also be used as a lemma in any general method. This is done by transforming any polygonal shape into a copy of itself but the area should be of any other given polygonal shape. 

Parbelos, a figure similar to an arbelos, doesn’t use semicircles. Instead, it makes use of parabola segments.